That's totally a tangent but I was reading a bit and it happens that, in practice, real cables bend in an inbetween curve between catenaries and parabolas
Comparison of a catenary arch (black dotted curve) and a parabolic arch (red solid curve) with the same span and sag. The catenary represents the profile of a simple suspension bridge, or the cable of a suspended-deck suspension bridge on which its deck and hangers have negligible weight compared to its cable. The parabola represents the profile of the cable of a suspended-deck suspension bridge on which its cable and hangers have negligible weight compared to its deck. The profile of the cable of a real suspension bridge with the same span and sag lies between the two curves. The catenary and parabola equations are respectively, y = cosh x and y = x²( (cosh 1) − 1) + 1
If the chain is carrying nothing other than its own weight, the resulting shape is a "catenary". If the chain is like a suspended cable carrying a deck below it, and its own weight is nothing compared to that of the deck, the resulting shape is a "parabola".
Which shows that sometimes your model (either using pure math or a simulation) is too simple to capture whatever is going on in the real world. (it gets further complicated when one considers elasticity etc)
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That's totally a tangent but I was reading a bit and it happens that, in practice, real cables bend in an inbetween curve between catenaries and parabolas
https://en.wikipedia.org/wiki/Catenary#Catenary_bridges
https://www.quora.com/How-do-you-tell-the-difference-between...
Which shows that sometimes your model (either using pure math or a simulation) is too simple to capture whatever is going on in the real world. (it gets further complicated when one considers elasticity etc)